1. Introduction
Josephson effect is one of the key components of macroscopic quantum phenomena, which occurs when a barrier (i.e. thin insulating tunnel barrier, ferromagnet, semiconductor, normal metal, etc) is placed between two superconductors. Quantum mechanical devices used for this purpose are Josephson junctions (JJs) [1–12]. The JJs have different applications in quantum mechanical and electro-opto-mechanical (EOM) circuits, such as superconducting quantum interference devices (SQUIDs) and superconducting qubits [1–13]. On the other hand, continuous quantum variables (CVs) are used to do quantum information processing tasks in quantum mechanical devices [14], that Gaussian states play an important role because they can be quickly generated from valid sources and controlled experimentally utilizing reliable sets of operations such as phase shifters, beam splitters, squeezers, and efficient detection systems [14]. They can be formed in different physical systems, such as light field modes [15, 16], quantum illumination [17, 18], quantum Lidar [19, 20], cold atoms [21], and excitons in photonic cavities [22]. Entanglement between two Gaussian modes is now produced experimentally in the laboratory, e.g. two output beams of a parametric down converter (signal/idler) are transmitted through optical fibers [14, 19, 20], or in atomic ensembles interacting with light [23] and two output beams of a Josephson parametric amplifiers (JPA) [24–29]. Such CV entanglement can be routinely generated utilizing squeezed light and non-linear optics [15, 16]. Two-mode entangled states increase the ability of two parties to communicate as well as other applications. The Gaussian states, also, are used in quantum key distributions [30], teleportation [31], and electromagnetically induced transparency [32].
In this study, due to the high importance of quantum circuits (JPA or EOM chip) in quantum radar and illumination, the application of entanglement, squeezing, and entropy behaviors in these circuits are investigated. Therefore, methods for improving the performance of these circuits and finally quantum radar are presented according to the study of their qualitative behaviors.
The structure of this article is as follows. The second part of the article introduces the basic principles of the QI, the third part introduces the model and the results of this work are expressed in section 4 .
2. Background
2.1. The principles of quantum radar/illumination
The principles of the QI operation can be expressed in four steps (figure 1) as follows [13, 33–52]:
| 1. | 1. Produce a current of entangled photon pairs (signal/idler). |
| 2. | 2. Amplify the signal, and record the idler. |
| 3. | 3. After receiving the reflected signal, amplify the signal and idler again and apply a match-filtering between the received signal and the previously recorded idler. |
| 4. | 4. Finally, the presence or absence of a target can be inferred by a suitable detector. |
Figure 1. Schematic representation of the quantum illumination (or radar) in general. |
The main idea of quantum two modes squeezed (QTMS) radar is originally derived from quantum illumination. In quantum illumination, signal delay lines and joint measurements between the reflected signal from the target and recorded idler are used, but in the QTMS radar inferred from QI, the signal and idler with time delay (due to the length of the free-space path of the transmitted signal) are measured at different times (does not require joint measurement and delay lines) [13, 33, 34, 37].
2.2. Two photons entanglement
Since the photons of the signal and idler originate from the same pump, there is a strong quantum correlation between two beams (the signal and idler), resulting in the squeezing of conjugate signals I and Q [13, 33–36, 38, 39, 41, 48]. The tool that is used to show this correlation between different system parameters, is its covariance matrix.
The root of entanglement comes back to the principle of quantum superposition and has no classical counterpart [53–57]. Consider a total system that has several bipartite composite systems (qubit). In general, in a quantum state, if the measurement on the first qubit (photon) affects the result of the measurement of the second qubit, we have an entangled state. Otherwise, it is non-entangled [54, 57]. It means that the quantum state of each particle of the pair or a group of particles cannot be described independently of the state of others.
There are different types of entanglement, for example, polarization entanglement [40, 52], number of photons [52], superconductivity (JJs) [13, 33–36, 38, 39, 41, 48], squeezed light [13, 33–36, 38, 39, 41], quantum dot [59, 60] and etc. Also, there are two types of CV entanglement such as voltage and discrete variable (DV) such as polarization [13, 33–36, 38, 39, 41, 48]. The entanglement of the CV of light squeezed by the JJ in the QTMS radars is used [33–36].
2.3. Quantum two-mode squeezed (QTMS) state
The JPAs can be used as devices that generate two-modes squeezed vacuum (-) signals [13, 33–36, 38, 39, 41]. The term vacuum refers to zero-point energy-related fluctuations, and amplifying these fluctuations is intended here and these fluctuations are not similar in the classic [33, 36]. In an absolute vacuum for quantum fields, the particles can fluctuate, and these fluctuations lead to the spontaneous emission from the excited state at the energy level E2 to the ground state at the energy level E1, and finally, it emits the energy difference between these two levels in the photons form with energy E2 − E1 = h&ngr; [13, 33–36, 38, 39, 41, 44].
3. The model
Consider two circuits as shown in figures 2(a) and (b). The first circuit is JPA [13, 24, 29], which we use in QTMS radar, figure 2(a). To perform radio detection and ranging of the QTMS, the JPA needs to be cooled at 7 mK by a dilution refrigerator. This temperature is required due to the presence of the SQUID in the JPA so that the entanglement between the signal and idler is not immediately eliminated by thermal noise [44]. The JPA used in this study is conventional and degenerate type i.e. has the same signal and idler frequencies ωi ≈ ωs = 5.31 GHz. The term of degenerate refers to the same frequency of the signal and idler in the JPA [13, 24, 29]. On the other hand, in figure 2(b), the EOM circuit as a source of two-mode squeezed thermal (TMST) [17, 18] that we use in QI is presented. Then, we consider two cases (1) the QTMS radar inferred from QI with JPA source, and (2) the QI with EOM source. The EOM source is used here has the same signal and idler frequencies ωi ≈ ωs = 5.31 GHz.
Figure 2. Schematic representation of the (a) JPA circuit and (b) EOM circuit. |
4. Results and discussion
4.1. Covariance matrix for two-mode Gaussian state
In general, the Hamiltonian of two-mode photons is equal to [53, 54]:
$\begin{eqnarray}H={\rm{i}}{\hslash }\left({{ga}}_{1}^{\dagger }{a}_{2}^{\dagger }-{g}^{* }{a}_{1}{a}_{2}\right),\end{eqnarray}$
where, subtitles 1 and 2 indicate modes, g is the coupling coefficient and a and a† are the photon annihilation and creation operators, respectively. In addition, ℏ = h/2π, h is Planck's constant. The two-mode squeezing operator is also expressed as follows [53, 54]: $\begin{eqnarray}S(\xi )=\exp \left(-{\xi }^{* }{a}_{1}{a}_{2}+\xi {a}_{1}^{\dagger }{a}_{2}^{\dagger }\right),\end{eqnarray}$
where, $\xi =r\exp ({\rm{i}}\phi )$ is an arbitrary complex number; r and φ are the amplitude (squeezing parameter) and the phase (squeezing angle), respectively. It must be noted that we do not consider the phase of the squeezing parameter.By acting squeezing operator on the photon annihilation and creation operators in the Hamiltonian and using the Baker–Campbell–Hausdorff formula, the normal shape of the covariance matrix for the quadratures of the resonators outputs is as follows [13, 17, 18, 33–36, 38, 39, 41, 43, 61–63]:
$\begin{eqnarray}C=\left[\begin{array}{cccc}{C}_{11} & 0 & {C}_{13} & 0\\ 0 & {C}_{11} & 0 & -{C}_{13}\\ {C}_{13} & 0 & {C}_{33} & 0\\ 0 & -{C}_{13} & 0 & {C}_{33}\end{array}\right].\end{eqnarray}$
Among various tools for measuring entanglement, one of the most appropriate criteria is logarithmic negativity, which is evaluated here [57]. The logarithmic negativity is defined by the following equation [17, 18, 63–74]: $\begin{eqnarray}{E}_{N}=\max \left[0,-\mathrm{log}(2{\eta }^{-})\right],\end{eqnarray}$
where η− is the smallest partially-transposed symplectic eigenvalue of C, and given by [17, 18, 59, 68, 72, 74]: $\begin{eqnarray}\begin{array}{l}{\eta }^{-}={2}^{-1/2}\\ \times \,{\left({C}_{11}^{2}+{C}_{33}^{2}+{C}_{13}^{2}-\sqrt{{\left({C}_{11}^{2}-{C}_{33}^{2}\right)}^{2}+4{C}_{13}^{2}{\left({C}_{11}-{C}_{33}\right)}^{2}}\right)}^{1/2},\end{array}\end{eqnarray}$
which, we have $\begin{eqnarray}\begin{array}{rcl}{C}_{11} & = & {C}_{22}=\left(\left(1+{n}_{1}+{n}_{2}\right)\cosh (2r)+\left({n}_{1}-{n}_{2}\right)\right)/2,\\ {C}_{33} & = & {C}_{44}=\left(\left(1+{n}_{1}+{n}_{2}\right)\cosh (2r)-\left({n}_{1}-{n}_{2}\right)\right)/2,\\ {C}_{13} & = & -{C}_{24}=\left(\left(1+{n}_{1}+{n}_{2}\right)\sinh (2r)+\cos (\phi )\right)/2.\end{array}\end{eqnarray}$
The average number of photons in the thermal noise in bath (a measure of the background noise power), is expressed as [46]: $\begin{eqnarray}{n}_{i}=1/\left(\exp (\hslash {\bar{\omega }}_{i}/{k}_{{\rm{B}}}T)-1\right),\end{eqnarray}$
where kB is Boltzmann constant, ${\bar{\omega }}_{i}=2\pi {\omega }_{i}$ and ωi is the signal and idler frequency (i = 1, 2) and T is the absolute temperature [68]. When ni = 0, the Gaussian state is called the TMSV [68]. In this case, when φ = 0 we have [62, 64]: $\begin{eqnarray}\begin{array}{rcl}{C}_{11} & = & {C}_{22}=\cosh (2r)/2,\\ {C}_{33} & = & {C}_{44}=\cosh (2r)/2,\\ {C}_{13} & = & -{C}_{24}=\sinh (2r)/2.\end{array}\end{eqnarray}$
When φ ≠ 0 we have [33–36, 39, 41, 62–64]: $\begin{eqnarray}\begin{array}{rcl}{C}_{11} & = & {C}_{22}=\cosh (2r)/2,\\ {C}_{33} & = & {C}_{44}=\cosh (2r)/2,\\ {C}_{13} & = & -{C}_{24}=\left(\sinh (2r)\cos (\phi )\right)/2.\end{array}\end{eqnarray}$
The squeezing parameter in quantum radar controls the power of the generated signal and idler, as well as the correlations between the signal and idler [62]. With increasing the squeezing parameter, the correlation between signal and idler increases, the power of the generated signal and idler increases, and the effect of the quantum noise decreases exponentially [62]. It means that the variances decrease and the noise in quadrature components are squeezed [62]. In the ideal case, the output power of the modes, and the covariance between them, depend on the squeezing parameter r, which is generally a monotonically increasing function of pump power [33]. Hence, the squeezing parameter is a significant key in the enhancement of the quantum radar performance.According to equations (3 )–(9 ), the qualitative behavior of entanglement versus temperature and squeezing parameter are plotted in figures 3(a)–(d). Figure 3(a), shows the entanglement versus temperature for various squeezing parameters. Figure 3(b), illustrates the entanglement versus squeezing parameter for various temperatures. In figure 3(c), the entanglement is shown in terms of squeezing parameter and temperature. Figure 3(d) confirms that by increasing both parameters (r and T) at the same time, the entanglement is maintained. The physical reason for this behavior is that maintaining entanglement requires noise and loss elimination, meaning that we must have a more correlated signal and idler to be able to maintain entanglement. Also, in figures 3(a)–(d) we see that, with increasing the squeezing parameter, the correlation between signal and idler increases, the effect of the quantum noise decreases, but the power of the generated signal and idler increases.
Figure 3. The EN entanglement behavior in terms of (a) temperature T (K) for different squeezing parameters (b) the squeezing parameter r at different temperatures (c) squeezing parameter for different squeezing angles (d) temperature and squeezing parameter. |
The qualitative behavior of entanglement versus temperature and squeezing angle are plotted in figure 4. Figure 4(a) shows the entanglement versus temperature for the various squeezing angles. Figure 4(b), illustrates the entanglement versus squeezing angle for the various temperatures. In figure 4(c), the entanglement in terms of the squeezing angle for the various squeezing parameters is shown. These three figures illustrate the need to zero consideration of the squeezing angle, which means that decreasing the squeezing angle φ can eliminate the system noise, loss and phase shift between transmitted signal and recorded idler [49, 62]. When φ = 0, the entanglement is maximum. In figure 4(c), also we see that with consideration of φ = 0 and increasing the squeezing parameter, the entanglement at room temperature is maintained.
Figure 4. The EN entanglement behavior versus (a) temperature for different squeezing angles (b) squeezing angle φ at different temperatures (c) squeezing angle for various squeezing parameters. |
4.1.1. Covariance matrix and entanglement of QTMS Radar
We consider a thermal bath with a large amount of thermal noise that contains the target, but there is no loss/noise (in the recorded idler) and no phase shift between the transmitted signal and recorded idler. Hence, φ = 0 (no system noise, no loss, and no phase shift), the elements of the covariance matrix for the QTMS radar are obtained [49, 62]:10 ), In figure 5, the qualitative behavior of the entanglement is plotted using the logarithmic negativity tool in terms of two quantities of temperature and squeezing parameter. As we can see, to survive the entanglement at high temperatures in the QTMS radar, the squeezing parameter should be considered very small, i.e. r < 0.7. Therefore, both entanglement and quantum correlation can be maintained in QTMS radar. In figures 5(a)–(c), unlike the TMST state, the entanglement in QTMS radar with the increasing of the squeezing parameter, the entanglement suppresses, the correlation increases, and the power of the generated signal and idler increases. The remarkable thing about this type of radar is that when the correlation and power are low, the entanglement is maximum. Also, at high temperature T > 400 mK, the entanglement survives when the squeezing parameter is low.
$\begin{eqnarray}\begin{array}{rcl}{C}_{11} & = & {C}_{22}=\left(\sqrt{\kappa }\left(\cosh (2r)-1\right)+2{n}_{i}+1\right)/4,\\ {C}_{33} & = & {C}_{44}=\cosh (2r)/4,\\ {C}_{13} & = & -{C}_{24}=\sqrt{\kappa }\sinh (2r)/4.\end{array}\end{eqnarray}$
The factor κ is the proportion of received photons in the receiver that originated in the QTMS radar. With respect to equation (
Figure 5. The EQN entanglement behavior in terms of (a) temperature T (K) for different squeezing parameters, (b) the squeezing parameter r at different temperatures, (c) temperature and squeezing parameter. |
4.2. Squeezing
The squeezing in the two-mode squeezed thermal state can be determined by the following expression [18]:
$\begin{eqnarray}\begin{array}{rcl}{S}^{Q}(0) & = & {C}_{11}+{C}_{33}-2{C}_{13}=(1+{n}_{1}+{n}_{2})\\ & & \times \left(\cosh (2r)-\sinh (2r)\cos (\phi )\right)/2.\end{array}\end{eqnarray}$
For φ = 0 and ni = 0 (TMSV), the squeezing will be $S(0)=\exp (-2r)/2$ .
The qualitative behavior of squeezing in terms of temperature and squeezing parameter are plotted in figure 6. Figure 6(a) shows the squeezing versus temperature for the various squeezing parameters. Figure 6(b), illustrates the squeezing versus squeezing parameter for the various temperatures. In figure 6(c), the squeezing versus squeezing parameter for various squeezing angles is shown. Figures 6(a)–(d) show that the qualitative behavior of squeezing when we increase temperature and squeezing parameter at the same time decreases (when φ = 0), but on the other hand, when φ ≠ 0, it gives the opposite result.
Figure 6. The S squeezing behavior as (a) temperature T (K) for different squeezing parameters (b) the squeezing parameter r at different temperatures (c) squeezing parameter with different squeezing angle and (d) squeezing parameter and temperature. |
The qualitative behavior of squeezing in terms of temperature and squeezing angle is plotted in figure 7. In figure 7(a), the squeezing behavior versus temperature for various squeezing angles is shown. Figure 7(b), illustrates the squeezing versus squeezing angle for various temperatures. In figure 7(c), the squeezing versus squeezing angle for various squeezing parameters is shown. We see that in figures 7(a)–(c), by increasing the temperature, squeezing angle, and squeezing parameter at the same time, the squeezing qualitative behavior increase.
Figure 7. The S squeezing behavior versus (a) temperature for different squeezing angles (b) squeezing angle φ at different temperatures (c) squeezing angle φ at different squeezing angles. |
4.2.1. Squeezing in the QTMS radar
The squeezing in the QTMS radar with respect to equation (11 ) [18], can be calculated by the squeezing formula as:
$\begin{eqnarray}\begin{array}{l}{S}^{Q}(0)={C}_{11}+{C}_{33}-2{C}_{13}\\ \,=\,{E}\left(\left.\kappa (\cosh (2r)-1)+2{n}_{i}+1+\cosh (2r\right)\right.\\ \,\left.-2\left(\left.\sqrt{\kappa }\sinh (2r\right)\right)\right)/4.\end{array}\end{eqnarray}$
The qualitative behavior of squeezing in terms of temperature and squeezing parameter, using equation (12 ), are plotted in figure 8. In figure 8(a), the squeezing versus temperature for various squeezing parameters is shown. Figure 8(b), illustrates the squeezing behavior versus squeezing parameter for various temperatures. In figure 8(c), the squeezing behavior versus squeezing parameter and temperature is shown. We see that the qualitative behavior of squeezing increases with the increase of temperature and the squeezing parameter in our QTMS radar.
Figure 8. The SQ squeezing behavior versus (a) temperature T (K) for different squeezing parameters, (b) the squeezing parameter r at different temperatures, (c) squeezing parameter and temperature. |
4.3. Entropy of formation
The entropy of formation and the effective number of ebits at the detectors input can be expressed in terms of the log-negativity defined in equation (4 ) as follow [18]:
$\begin{eqnarray}{E}_{f}={\sigma }_{+}{\mathrm{log}}_{2}{\sigma }_{+}-{\sigma }_{-}{\mathrm{log}}_{2}{\sigma }_{-},\end{eqnarray}$
where ${\sigma }_{\pm }={\left((1/\sqrt{\theta })\pm \sqrt{\theta }\right)}^{2}/4$ with $\theta ={2}^{-{E}_{N}}$.The comparison of the qualitative behavior of squeezing, entanglement, and entropy of formation in terms of temperature, squeezing parameter, and squeezing angle is plotted in figure 9. Figures 9(a)–(c) illustrate the qualitative behaviors of entanglement and entropy are similar, but the behavior of squeezing is the opposite. It means that the behavior of the effective number of ebits at the detectors input corresponds to the behavior of entanglement.
Figure 9. The comparison behaviors of squeezing S, entanglement EN and entropy Ef versus (a) temperature T (K) (b) the squeezing parameter r, (c) squeezing angle φ. |
4.3.1. Entropy of formation in the QTMS radar
The comparison plots of the qualitative behavior of squeezing, entanglement, and entropy of formation (for QTMS state) in terms of squeezing parameter, and temperature are shown in figure 10. It shows that the qualitative behavior of entanglement and entropy are similar for QTMS state. When the squeezing parameter is very low i.e. r < 0.7, the entanglement maintains (see figure 10(a)), but with increasing the squeezing parameter, the entanglement is suppressed, and squeezing increases. We see that in figure 10(b), also, the entanglement proceeds in a straight line with increasing temperature. Even at room temperature, the entanglement is maintained but is low. However, the qualitative behavior of squeezing at room temperature is maximum.
Figure 10. The comparison behaviors of squeezing SQ, entanglement EQN and entropy EQf versus (a) squeezing parameter r (b) temperature T(K). |
4.4. Comparison of TMSV, TMST and QTMS states
The qualitative behavior comparison of squeezing, and entanglement, in terms of squeezing parameter for TMSV state, TMST state, and QTMS radar at room temperature are plotted in figures 11(a) and (b), respectively. In this figure, we see that the behavior of squeezing and entanglement in the TMSV and TMST are opposite for QTMS radar. It is clear that in figure 11(b), the entanglement in the QTMS radar for particular conditions (i.e. r < 0.8 and φ = 0) is maintained. However, in the TMST state, the entanglement increases with increasing the squeezing parameter.
Figure 11. The qualitative behavior comparison of (a) squeezing S, (b) entanglement EN. |
The qualitative behavior comparison of entanglement in terms of temperature in the TMST state and QTMS radar are plotted in figures 12(a) and (b). The entanglement in the QTMS radar for particular conditions (figure 12(a) r = 0, φ = 0) is maintained at room temperature, but in other conditions (figure 12(b) r = 1, φ = 0) the entanglement in QTMS radar is suppressed for T > 0.3 K. It is clear that entanglement in the QTMS radar survives at room temperature when the squeezing parameter is low 0 < r < 0.7, i.e. when power of the generated signal and idler is low.
Figure 12. The qualitative behavior comparison of entanglement EN in terms of temperature T(K) between TMST and QTMS states (a) r = 0 , (b) r = 1. |
5. Conclusion
In this study, the qualitative behaviors of entanglement, squeezing, and entropy in QTMS radar with a JPA circuit that generates the QTMS state, and QI with an EOM circuit that generates the TMST state were investigated. For the first time, we calculated the entanglement, squeezing and entropy for the QTMS radar when the target is present and the signal is transmitted to the target. In addition, by controlling the squeezing parameter which is a tool to control entanglement, entropy, and squeezing, the performance of the QTMS radar can be improved, so this work showed how it is implemented in practice. The qualitative behaviors of entanglement and entropy are quite similar in the two cases. It means that the behavior of the effective number of ebits at the detectors input corresponds to the behavior of entanglement. In QTMS radars with the JPA source, we found that the entanglement does not disappear. We can also see that the entanglement is maintained in this type of radar by changing the temperature and reducing the squeezing parameter. The remarkable thing about QTMS radar is that when the correlation and power of the generated signal and idler are low, the entanglement is maximum. And, at high temperature T > 400 mK, the entanglement survives when the squeezing parameter is low. It means that the QTMS radars are interested in the low-power regime. In the QI with an EOM source, we have shown that the entanglement is maintained by increasing the squeezing parameter, which means that, in this protocol, we need the high correlated signal and idler to survive entanglement. The significant result obtained from the qualitative behavior of the QI was that at high power of the signal and idler generation, low temperature and high correlation, the entanglement increases. On the other hand, the qualitative behavior of the squeezing increases in the QTMS radar, when increasing the temperature and squeezing parameter. But, under the same condition, the qualitative behavior of the squeezing in the TMST state decreases when increasing the temperature and the squeezing parameter. Finally, it should be noted that the squeezing parameter plays a very important role in the performance of quantum radar and quantum illumination, which must be taken into account both in theoretical studies and in the design of quantum radar and illumination.
Acknowledgments
The authors would like to thank J Teixeira for helping with the manuscript and valuable comments.
Author contributions
The authors SMH and MN contributed equally to this work.